We determine thresholds $p_c$ for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The proposed single scalar index $\xi ={\sum }_i{z}_i{r}_i^2/i$ (depending on the coordination zone number $i$, the neighborhood coordination number $z$, and the square distance $r^2$ to sites in $i\mathrm{th}$ coordination zone from the central site) allows one to differentiate among various neighborhoods and relate $p_c$ to $\xi$. The thresholds roughly follow a power law $p_c\propto {\xi }^{-\gamma }$ with $\gamma \approx 0.710\left(19\right)$.

• Received 24 February 2021
• Accepted 12 April 2021

DOI:https://doi.org/10.1103/PhysRevE.103.052107